| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Linear combinations of independent variables |
| Difficulty | Moderate -0.3 This is a straightforward application of standard formulas for expectation and variance. Part (a) requires basic calculations from a probability distribution (E(R) = Σrp(r) and Var(R) = E(R²) - [E(R)]²). Part (b) applies linear combination formulas including the correlation term, which is A-level content but routine once the formulas are known. The calculations are mechanical with no problem-solving or conceptual challenges beyond formula recall. |
| Spec | 5.02b Expectation and variance: discrete random variables5.04b Linear combinations: of normal distributions |
| \(\boldsymbol { r }\) | 6 | 7 | 8 |
| \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\) | 0.1 | 0.6 | 0.3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(R) = (6 \times 0.1)+(7 \times 0.6)+(8 \times 0.3) = 0.6+4.2+2.4 = 7.2\) | B1 | CAO |
| \(E(R^2) = (3.6+29.4+19.2) = 52.2\) | B1 | CAO |
| \(\text{Var}(R) = E(R^2) - (E(R))^2\) | M1 | Use of formula |
| \(= 52.2 - 51.84 = 0.36\) | A1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(T) = 7.2 + 10.9 = 18.1\) | B1\(\checkmark\) | \(\checkmark\) on \(E(R)\) |
| \(\text{Cov}(R,S) = \rho_{RS} \times \sqrt{\text{Var}(R) \times \text{Var}(S)}\) | M1 | Use of; or equivalent. May be scored in (ii) |
| \(\text{Var}(T) = \text{Var}(R) + \text{Var}(S) + 2\text{Cov}(R,S)\) | M1 | Use of; or equivalent. May be scored in (ii) |
| \(= 0.36 + 1.69 + 2 \times \frac{2}{3}\sqrt{0.36 \times 1.69}\) | ||
| \(= 0.36 + 1.69 + 1.04 = 3.09\) | A1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(D) = 10.9 - 7.2 = 3.7\) | B1\(\checkmark\) | \(\checkmark\) on \(E(R)\) |
| \(\text{Var}(D) = \text{Var}(S) + \text{Var}(R) - 2\text{Cov}(S,R)\) | ||
| \(= 1.69 + 0.36 - 2 \times \frac{2}{3}\sqrt{1.69 \times 0.36}\) | ||
| \(= 1.69 + 0.36 - 1.04 = 1.01\) | B1 | CAO |
# Question 4:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(R) = (6 \times 0.1)+(7 \times 0.6)+(8 \times 0.3) = 0.6+4.2+2.4 = 7.2$ | B1 | CAO |
| $E(R^2) = (3.6+29.4+19.2) = 52.2$ | B1 | CAO |
| $\text{Var}(R) = E(R^2) - (E(R))^2$ | M1 | Use of formula |
| $= 52.2 - 51.84 = 0.36$ | A1 | CAO |
## Part (b)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(T) = 7.2 + 10.9 = 18.1$ | B1$\checkmark$ | $\checkmark$ on $E(R)$ |
| $\text{Cov}(R,S) = \rho_{RS} \times \sqrt{\text{Var}(R) \times \text{Var}(S)}$ | M1 | Use of; or equivalent. May be scored in (ii) |
| $\text{Var}(T) = \text{Var}(R) + \text{Var}(S) + 2\text{Cov}(R,S)$ | M1 | Use of; or equivalent. May be scored in (ii) |
| $= 0.36 + 1.69 + 2 \times \frac{2}{3}\sqrt{0.36 \times 1.69}$ | | |
| $= 0.36 + 1.69 + 1.04 = 3.09$ | A1 | CAO |
## Part (b)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(D) = 10.9 - 7.2 = 3.7$ | B1$\checkmark$ | $\checkmark$ on $E(R)$ |
| $\text{Var}(D) = \text{Var}(S) + \text{Var}(R) - 2\text{Cov}(S,R)$ | | |
| $= 1.69 + 0.36 - 2 \times \frac{2}{3}\sqrt{1.69 \times 0.36}$ | | |
| $= 1.69 + 0.36 - 1.04 = 1.01$ | B1 | CAO |4 The table below shows the probability distribution for the number of students, $R$, attending classes for a particular mathematics module.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$\boldsymbol { r }$ & 6 & 7 & 8 \\
\hline
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$ & 0.1 & 0.6 & 0.3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find values for $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$.
\item The number of students, $S$, attending classes for a different mathematics module is such that
$$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$
Find values for the mean and variance of:
\begin{enumerate}[label=(\roman*)]
\item $T = R + S$;
\item $\quad D = S - R$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S3 2006 Q4 [6]}}